Forecasting Bitcoin Prices: Insights and Analysis

CONTRIBUTORS

Noor A Tanjum Saba Amin (241000161)
A N M Zahid Hossain (241002061)
Kazi Nabila Tasnim (241001961)
Upam Chowdhury (241001161)
Mohammad Shafiur Rahman (241000661)

INTRODUCTION

This project focuses on predicting Bitcoin prices using Regression and Time Series analysis, a critical area in financial time series research due to Bitcoin’s volatility and market significance. Regression analysis, including linear and quadratic models, forms the backbone of the study, allowing for the identification of trends and relationships in Bitcoin prices. While linear regression offers simplicity and interpretability, quadratic regression captures non-linear patterns more effectively. Time Series analysis, particularly using ARIMA (AutoRegressive Integrated Moving Average) models, is also pivotal. ARIMA models are adept at handling temporal dependencies and making data stationary, essential for accurate forecasting. Studies have shown ARIMA’s efficacy in financial markets, emphasizing its utility in predicting cryptocurrency prices. A comparative analysis of different models—assessed using metrics like , BIC, and RMSE—ensures the selection of the most suitable approach. Previous research, highlights that model performance varies with datasets, underscoring the importance of this comparative approach. This project not only applies these robust methodologies to Bitcoin but also aims to inform investment strategies and policy decisions by providing accurate forecasts. The integration of multiple modeling approaches—linear, quadratic, and ARIMA—enhances the robustness of the predictions, making a significant contribution to the field of financial time series analysis and the practical domain of cryptocurrency market forecasting.

BACKGROUND & SIGNIFICANCE

Bitcoin, as the pioneer cryptocurrency, has exhibited substantial volatility and growth since its inception. Understanding its price dynamics is crucial for investors, policymakers, and researchers. Previous studies have employed various statistical and machine learning models to forecast Bitcoin prices, emphasizing the importance of accurate predictive models in financial markets.

4.1 Loading the Dataset

Load the dataset from csv into a data frame named BitCoin.

# Load necessary library
library(tidyverse)
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# Load the dataset
BitCoin <- read.csv("C:/Users/chowd/Downloads/BitCoin.csv")

# Check the first few rows of the dataset
head(BitCoin)
##         Date   Close
## 1 2015-01-01 217.464
## 2 2015-02-01 254.263
## 3 2015-03-01 244.224
## 4 2015-04-01 236.145
## 5 2015-05-01 230.190
## 6 2015-06-01 263.072

Check the data types of the features

# Check the data types of the features
str(BitCoin)
## 'data.frame':    107 obs. of  2 variables:
##  $ Date : chr  "2015-01-01" "2015-02-01" "2015-03-01" "2015-04-01" ...
##  $ Close: num  217 254 244 236 230 ...

Assign appropriate data type to features.

# The date column is not in Date format, So Now we convert it into Date format
BitCoin$Date <- as.Date(BitCoin$Date, format = "%Y-%m-%d")

# Check the structure again to confirm changes
str(BitCoin)
## 'data.frame':    107 obs. of  2 variables:
##  $ Date : Date, format: "2015-01-01" "2015-02-01" ...
##  $ Close: num  217 254 244 236 230 ...

Check the structure of the data frame

# Check the structure of the data frame
summary(BitCoin)
##       Date                Close        
##  Min.   :2015-01-01   Min.   :  217.5  
##  1st Qu.:2017-03-16   1st Qu.: 1263.9  
##  Median :2019-06-01   Median : 8658.5  
##  Mean   :2019-06-01   Mean   :14944.3  
##  3rd Qu.:2021-08-16   3rd Qu.:24634.2  
##  Max.   :2023-11-01   Max.   :61319.0

Check if there are any missing values and handle them appropriately.

# Check for missing values
sum(is.na(BitCoin))
## [1] 0

There is no missing value found.

4.2 Descriptive Analytics

Copy the BitCoin data frame to a new data frame named BitCoin_df. In the new data frame create two more columns ‘month’ & ‘year’ by populating with the months & years values from the ‘Date’ column.

# Copy the data frame to a new data frame named BitCoin_df
BitCoin_df <- BitCoin

# Create 'Month' and 'Year' columns
BitCoin_df <- BitCoin_df %>%
  mutate(Month = format(Date, "%m"),
         Year = format(Date, "%Y"))

# Check the structure of the new data frame
str(BitCoin_df)
## 'data.frame':    107 obs. of  4 variables:
##  $ Date : Date, format: "2015-01-01" "2015-02-01" ...
##  $ Close: num  217 254 244 236 230 ...
##  $ Month: chr  "01" "02" "03" "04" ...
##  $ Year : chr  "2015" "2015" "2015" "2015" ...

Create a monthly boxplot of prices.

# Create a monthly boxplot of prices 
ggplot(BitCoin_df, aes(x = Month, y = Close)) +
  geom_boxplot() +
  labs(title = "Monthly Boxplot of Bitcoin Prices", x = "Month", y = "Price") +
  theme_minimal()

Create a yearly boxplot of prices.

# Create a yearly boxplot of prices 
ggplot(BitCoin_df, aes(x = Year, y = Close)) +
  geom_boxplot() +
  labs(title = "Yearly Boxplot of Bitcoin Prices", x = "Year", y = "Price") +
  theme_minimal()

Create year wise trend lines of prices.

library(plotly)
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# Create year-wise trend lines of prices 
plot_ly(BitCoin_df, x = ~Date, y = ~Close, color = ~Year, type = 'scatter', mode = 'lines') %>%
  layout(title = "Year-wise Trend Lines of Bitcoin Prices",
         xaxis = list(title = "Year"),
         yaxis = list(title = "Price"))
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Convert the BitCoin data frame to a time series object with frequency 1:

library(xts)
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# Convert to xts object
BitCoin_xts <- xts(BitCoin_df$Close, order.by = BitCoin_df$Date)

# Convert BitCoin_xts to data frame for further analysis or plotting
BitCoin_dfm<- data.frame(Date = index(BitCoin_xts), Close = coredata(BitCoin_xts))

Plot the time series of monthly prices on years:

# Plotting time series
plot(BitCoin_xts, type = "o", col = "blue", xlab = "Year", ylab = "Price (USD)",
     main = "Time Series of Monthly Bitcoin Prices")

Find the relationship between consecutive month and show correlation through a scatter plot:

library(ggplot2)
# Create a data frame with lagged values
BitCoin_lag <- BitCoin_df
BitCoin_lag$Close_lag <- c(NA, head(BitCoin_df$Close, -1))  # Create a lagged version of Close

# Remove the first row with NA
BitCoin_lag <- na.omit(BitCoin_lag)

# Scatter plot of closing prices vs lagged closing prices
ggplot(BitCoin_lag, aes(x = Close_lag, y = Close)) +
  geom_point(color = "blue") +
  labs(x = "Previous Month's Price (USD)", y = "Current Month's Price (USD)",
       title = "Scatter Plot of Consecutive Monthly Prices") +
  geom_smooth(method = "lm", col = "red")  # Add a linear regression line
## `geom_smooth()` using formula = 'y ~ x'

4.3 Regression Analysis

4.3.1 Linear Regression

Create a linear model of the time series dataset.

# Create linear model
lm_model <- lm(Close ~ Date, data = BitCoin)

Show the summary of the model and explain the outcome:

# Summary of linear model
summary(lm_model)
## 
## Call:
## lm(formula = Close ~ Date, data = BitCoin)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -15114  -7997  -2255   3065  35626 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -2.211e+05  1.939e+04  -11.40   <2e-16 ***
## Date         1.308e+01  1.073e+00   12.19   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 10430 on 105 degrees of freedom
## Multiple R-squared:  0.586,  Adjusted R-squared:  0.5821 
## F-statistic: 148.6 on 1 and 105 DF,  p-value: < 2.2e-16

Explanation:

The linear regression analysis where the Close price of Bitcoin is regressed on the Date reveals several key findings. The formula used in the regression model is Close ~ Date, indicating that the Close price of Bitcoin is modeled as a function of Date. The residuals, which represent the differences between the observed and predicted values, have a wide range, with the minimum residual at -15,114, the first quartile at -7,997, the median at -2,255, the third quartile at 3,065, and the maximum residual at 35,626. This range indicates some degree of variability in the model’s predictions.

The coefficients in the model include an intercept of -221,100 and a Date coefficient of 13.08. The intercept is the estimated value of the Close price when Date is zero. However, in the context of a time series, the intercept mainly serves as a starting point for the regression line. The Date coefficient suggests that, on average, the Bitcoin Close price increases by about $13.08 per unit of time. The standard error for the intercept is 19,390, and for the Date coefficient, it is 1.073. These standard errors measure the average distance that the observed values fall from the regression line, with smaller standard errors indicating more precise estimates.

The t-values and associated p-values for the intercept and Date coefficients are highly significant. The t-value for the intercept is -11.40 with a p-value less than 2e-16, and the t-value for the Date coefficient is 12.19 with a p-value also less than 2e-16. These small p-values indicate that both the intercept and the Date coefficient are statistically significant at conventional significance levels. The residual standard error of the model is 10,430 on 105 degrees of freedom, indicating the typical size of the residuals. A lower value would suggest a better fit of the model to the data.

The multiple R-squared value is 0.586, meaning that approximately 58.6% of the variability in the Close price of Bitcoin can be explained by the Date variable. The adjusted R-squared value is 0.5821, which is a modified version of R-squared that accounts for the number of predictors in the model. The F-statistic for the overall significance of the model is 148.6 on 1 and 105 degrees of freedom, with a p-value less than 2.2e-16, indicating that the model is statistically significant.

In summary, the linear regression model suggests a statistically significant upward trend in Bitcoin prices over time, with the Close price increasing by about $13.08 per unit of time. The model explains approximately 58.6% of the variability in Bitcoin prices, indicating a moderate fit. However, the large residual standard error suggests substantial variability not explained by the model, possibly due to other influencing factors not included in this simple linear model.

Create a plot of the linear model on top of the time series dataset line plot with scatter data points:

library(colorspace)
## Warning: package 'colorspace' was built under R version 4.3.3
library(plotly)

# Predicted values from linear model
BitCoin_df$Predicted <- predict(lm_model)

# Create a color palette with enough colors for all unique years
num_years <- length(unique(BitCoin_df$Year))
colors <- qualitative_hcl(num_years, palette = "Set2")

# Plot with Plotly
plot_ly(BitCoin_df, x = ~Date, y = ~Close, color = ~Year, colors = colors,
                type = 'scatter', mode = 'lines+markers', 
                marker = list(size = 6),
                line = list(width = 1), name = "Actual") %>%
  add_trace(x = ~Date, y = ~Predicted, mode = 'lines', 
            line = list(color = 'red', width = 2), name = "Predicted") %>%
  layout(title = "Linear Regression Model on Bitcoin Prices",
         xaxis = list(title = "Month"),
         yaxis = list(title = "Closing Price (USD)"),
         legend = list(title = list(text = "Year"),
                       traceorder = "reversed",
                       tracegroupgap = 10))
## A marker object has been specified, but markers is not in the mode
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Explanation:

The time series plot tracks the monthly closing prices of Bitcoin in US dollars, spanning January 2015 to December 2023 (assuming the data includes December 2023). Each data point on the plot represents an actual monthly closing price, and together they illustrate the price movements over this extended period.

Time Series and Trend: The x-axis acts as our time scale, displaying months with year labels appearing at every January tick mark. The y-axis represents the price of Bitcoin in US dollars. The line superimposed on these data points showcases the general trend in Bitcoin’s closing price over time.

Linear Regression Model: This line represents a statistical model attempting to capture the overall trend in the data. By fitting a straight line through the scatter plot of closing prices, the model offers a simplified representation of the average price movement. The specific equation for this line might be displayed elsewhere but isn’t visible in the image we’re examining.

Key Observations:

Upward Trajectory: By analyzing the time series plot, we can observe a general upward trend. This upward direction is mirrored by the positive slope of the linear regression model. This suggests that, on average, the closing price of Bitcoin has increased throughout the period from January 2015 to November 2023.

Price Fluctuations: Despite the apparent upward trend, the data points themselves reveal significant volatility in Bitcoin’s price. The scatter plot shows these data points spread out around the regression line, indicating that the price has fluctuated considerably from month to month. The linear regression model provides an average trend but doesn’t capture these individual price swings.

Limitations of the Model: It’s crucial to remember that linear regression models are simplifications of real-world phenomena. The price of Bitcoin is influenced by a multitude of complex factors, and a straight line may not perfectly capture the intricate price movements. The deviations of the scatter points from the regression line highlight this limitation.

In conclusion, the graph offers insights into Bitcoin’s price behavior. While the overall trend suggests a general increase over time, the data also demonstrates substantial volatility.

Perform residual analysis and create a line & scatter plot of the residuals:

# Load necessary packages
library(dplyr)
library(magrittr)
## 
## Attaching package: 'magrittr'
## The following object is masked from 'package:purrr':
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##     set_names
## The following object is masked from 'package:tidyr':
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library(plotly)
library(xts)


# Residual analysis
residuals <- residuals(lm_model)

# Create a data frame for residuals plot using the original Date column
residuals_df <- data.frame(Date = index(BitCoin_xts), Residuals = residuals)

# Plot residuals
plot_ly(residuals_df, x = ~Date, y = ~Residuals, 
        type = 'scatter', mode = 'markers',  # First trace with markers
        marker = list(color = 'blue', size = 4)) %>%
  add_trace(x = ~Date, y = ~rep(0, length(residuals_df$Date)), 
            type = 'scatter', mode = 'lines',  # Second trace with only line
            line = list(color = 'red', width = 6)) %>%
  layout(title = "Residuals Plot of Linear Regression Model",
         xaxis = list(title = "Date"),
         yaxis = list(title = "Residuals"))
## A marker object has been specified, but markers is not in the mode
## Adding markers to the mode...

Explanation:

The provided plot depicts residuals from a linear regression model, where residuals represent the differences between observed values and those predicted by the model. The X-axis denotes dates ranging from 2015 to 2024, while the Y-axis measures residuals. In the plot, blue dots (Trace 0) signify individual residual values, while a red line (Trace 1) serves as a trend line, ideally centered around zero.

Interpreting the plot reveals that residuals span from approximately -10,000 to +30,000, showing variability over time. Initially, from 2015 to around 2018, residuals cluster tightly around zero with minimal deviation. Post-2018, however, the spread of residuals noticeably widens. Around 2020, there’s a notable clustering of residuals below zero, while by 2022, variability increases further with some residuals showing notably high positive values.

Analyzing trends, the horizontal red line at zero indicates that residuals should ideally hover around zero, implying a well-fitted model. However, deviations from this line highlight periods where model predictions were less accurate. Potential issues identified include heteroscedasticity, evident from the increasing spread of residuals over time, suggesting that the variance of residuals isn’t constant. Moreover, the model’s fit appears to falter, especially noted by the pronounced spread of residuals in 2022, hinting at possible outliers or influential points impacting accuracy.

In conclusion, the residuals plot points towards limitations in the linear regression model’s ability to predict values consistently across different time periods. Addressing these challenges may require further investigation, potential model refinements, or the incorporation of more sophisticated variables to enhance prediction accuracy.

Create a histogram plot of the residuals:

# Histogram of residuals
plot_ly(residuals_df, x = ~Residuals, type = 'histogram') %>%
  layout(title = "Histogram of Residuals",
         xaxis = list(title = "Residuals"),
         yaxis = list(title = "Frequency"))

Explanation:

The provided histogram illustrates the distribution of residuals from a linear regression model, focusing on the differences between observed and predicted values. The X-axis represents the range of residuals, spanning from approximately -20,000 to +40,000, while the Y-axis shows the frequency of residuals within each bin.

Interpreting the histogram reveals that most residuals are centered around zero, indicating that the model’s predictions generally align closely with the observed data. A notable peak at zero suggests a significant number of residuals are very close to this value. However, the distribution exhibits a slight right skew, implying there are some larger positive residuals where the model underestimated the observed values. Fewer residuals appear in the positive range (especially above 20,000), contrasting with a visible number of residuals in the negative range, albeit less extreme.

Moreover, outliers are evident in the positive range of residuals, particularly around 30,000 to 40,000, highlighting instances where the model significantly mispredicted. Despite the majority of residuals being near zero, the histogram’s right skew and the presence of outliers suggest deviations from a perfectly normal distribution. This indicates potential data patterns or anomalies that the linear regression model may not fully capture.

In conclusion, while the linear regression model generally performs well with predictions close to observed values, the histogram suggests areas for improvement. Addressing the right skew, handling outliers, and potentially revising the model or exploring data transformations could enhance its accuracy and better accommodate underlying data nuances.

Create ACF & PACF plots of residuals:

# ACF plot of residuals
acf_res <- acf(residuals, main = "ACF of Residuals")

# PACF plot of residuals
pacf_res <- pacf(residuals, main = "PACF of Residuals")

Explanation:

ACF

This Autocorrelation Function (ACF) plot of residuals from a time series model provides insights into the correlation between residuals and their lagged values. The Y-axis represents the autocorrelation coefficient, measuring this relationship, while the X-axis denotes the number of time steps (lags) between compared residuals. Each vertical bar on the plot illustrates the autocorrelation coefficient for a specific lag, with bar height indicating the strength and direction of correlation. Additionally, blue dashed lines signify confidence intervals typically set at ±2/√n, where n is the sample size, helping identify statistically significant correlations.

Interpreting the plot reveals significant autocorrelations at initial lags, suggesting residual correlation that the model may not have fully captured. This implies potential inadequacies in the model’s ability to account for underlying data patterns. As lag increases, autocorrelations diminish and generally fall within the confidence intervals, indicating that residuals are less correlated at higher lags. Ideally, residuals should exhibit near-zero autocorrelation for all lags if the model is appropriate. Persistent significant autocorrelations imply possible model shortcomings, such as omitted variables or incorrect specifications.

In summary, this ACF plot highlights residual autocorrelation at lower lags, signaling areas where the model could benefit from refinement to better capture data dynamics. Addressing these findings may involve revisiting model assumptions, incorporating additional variables, or exploring alternative model specifications to improve overall predictive accuracy.

PACF

This Partial Autocorrelation Function (PACF) plot of residuals from a time series model provides insights into the partial correlation between residuals and their lagged values, accounting for intervening lags. The Y-axis represents the partial autocorrelation coefficient, which measures this relationship, while the X-axis denotes the number of time steps (lags) between compared residuals. Each vertical bar on the plot illustrates the partial autocorrelation coefficient for a specific lag, with bar height indicating the strength and direction of the partial correlation. Blue dashed lines represent confidence intervals typically set at ±2/√n, where n is the sample size, indicating statistically significant correlations.

Interpreting the plot reveals a significant partial autocorrelation at lag 1, suggesting a direct relationship between residuals and their first lag. This indicates that after accounting for other intervening lags, there remains a notable residual correlation at this lag. For lags beyond 1, the partial autocorrelations generally fall within the confidence intervals, implying no significant residual correlations once the first lag is considered. This pattern suggests that the model adequately captures the relationships between residuals and lagged values beyond the first.

Ideally, residuals should exhibit near-zero partial autocorrelation for all lags if the model is appropriate. The significant partial autocorrelation at lag 1 in this PACF plot suggests that the model might benefit from further refinement to better capture or incorporate a first-order component in its structure. Addressing this finding could involve reassessing model specifications, incorporating additional explanatory variables, or exploring alternative modeling approaches to improve predictive accuracy and account for residual correlations effectively.

Create QQ plot of residuals:

# QQ plot of residuals
qqnorm(residuals)
qqline(residuals)

Explanation:

Understanding the Plot:

The x-axis represents the theoretical quantiles, or expected values, of a normal distribution. The y-axis represents the quantiles of the residuals calculated from our linear regression model. In an ideal scenario, if the residuals follow a normal distribution, the data points on the plot would closely resemble a straight diagonal line. This would signify that the residuals behave similarly to what we would expect from a normal distribution.

Observations from the Data:

Examining the normal QQ plot we have, it appears that the data points deviate somewhat from the ideal straight diagonal line, particularly at the tails of the distribution (the areas on the far left and right sides of the plot). This suggests a potential departure from normality in the residuals.

Perform Shapiro-Wilk test on residuals:

# Shapiro-Wilk test on residuals
shapiro.test(residuals)
## 
##  Shapiro-Wilk normality test
## 
## data:  residuals
## W = 0.85983, p-value = 1.215e-08

Explanation:

The statement “Shapiro-Wilk normality test” followed by the results “data: residuals W = 0.85983, p-value = 1.215e-08” signifies the outcome of a statistical test used to assess whether a set of data, specifically the residuals from a statistical model, adheres to a normal distribution. The Shapiro-Wilk test evaluates the null hypothesis that the data is sampled from a population that follows a normal distribution. In this context, the test was applied to the residuals, which are the differences between observed values and those predicted by the model.

The test statistic, denoted as W, is calculated to quantify how closely the sample resembles a normal distribution. A value of W close to 1 suggests strong conformity to normality. Here, the computed W value is 0.85983, indicating some departure from normality. The critical component of the test is the p-value, which measures the probability of observing the data if the null hypothesis were true (i.e., if the residuals were normally distributed).

In this case, the p-value is very small, specifically 1.215e-08 (or 0.00000001215 in standard notation), which is significantly less than common significance levels like 0.05 or 0.01. This low p-value provides strong evidence against the null hypothesis of normality. Therefore, we reject the null hypothesis and conclude that the residuals do not follow a normal distribution.

The implication of non-normality in residuals is important for interpreting statistical models. Departure from normality can impact the accuracy of statistical inferences and predictions derived from the model. It suggests that the assumptions underlying the model (such as normality of errors) may not be fully satisfied. As a result, further investigation may be warranted to understand the nature of the deviation from normality, potentially requiring adjustments to the model or data transformations to improve its validity and reliability.

Explain if linear model is appropriate or not.

Based on the provided information and analysis of residuals, it appears that the linear regression model used to predict Bitcoin’s closing prices over time has significant limitations. The residuals have a wide range, spanning from -15,114 to 35,626, indicating substantial variability in the model’s predictions. The model’s coefficients, specifically an intercept of -221,100 and a Date coefficient of 13.08, are statistically significant but suggest the model’s simplicity might not capture the complexity of Bitcoin’s price movements. The multiple R-squared value of 0.586 indicates that about 58.6% of the variability in Bitcoin prices can be explained by the Date variable, but the residual standard error of 10,430 suggests large residuals and significant unexplained variability.

The residuals plot reveals patterns and increasing variability over time, indicating potential issues with the model and suggesting heteroscedasticity (non-constant variance of residuals). The histogram shows a right-skewed distribution with outliers, indicating that the residuals are not normally distributed. These findings violate key assumptions of linear regression, such as normality of residuals and constant variance. Additionally, while the model explains a moderate portion of the variability in Bitcoin prices, the large residual standard error and presence of outliers indicate that a simple linear model is insufficient to capture the complexities of Bitcoin price movements.

4.3.2 Quadratic Regression

Create a quadratic model of the time series dataset.

# Create quadratic model
BitCoin_df$Date_numeric <- as.numeric(BitCoin_df$Date)
quadratic_model <- lm(Close ~ poly(Date_numeric, 2, raw = TRUE), data = BitCoin_df)

Show the summary of the model and explain the outcome.

# Summary of quadratic model
summary(quadratic_model)
## 
## Call:
## lm(formula = Close ~ poly(Date_numeric, 2, raw = TRUE), data = BitCoin_df)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -15872  -7420  -1996   2666  36106 
## 
## Coefficients:
##                                      Estimate Std. Error t value Pr(>|t|)
## (Intercept)                         1.066e+05  4.157e+05   0.256    0.798
## poly(Date_numeric, 2, raw = TRUE)1 -2.333e+01  4.615e+01  -0.506    0.614
## poly(Date_numeric, 2, raw = TRUE)2  1.009e-03  1.278e-03   0.789    0.432
## 
## Residual standard error: 10450 on 104 degrees of freedom
## Multiple R-squared:  0.5885, Adjusted R-squared:  0.5806 
## F-statistic: 74.36 on 2 and 104 DF,  p-value: < 2.2e-16

Explanation:

The linear regression model aimed to predict Bitcoin’s closing prices using a second-degree polynomial of the numeric date. The model’s coefficients for the intercept, first-degree term, and second-degree term are not statistically significant, with p-values of 0.798, 0.614, and 0.432, respectively. Despite this, the model explains a considerable portion of the variance in the data, as indicated by an R-squared value of 0.5885 and an adjusted R-squared of 0.5806, suggesting that about 58% of the variability in Bitcoin’s closing prices is accounted for by the model. The residual standard error is 10,450, reflecting moderate dispersion of the residuals. The overall model is highly significant, with an F-statistic of 74.36 and a p-value less than 2.2e-16, indicating that the model provides a significant fit to the data as a whole.

Explain if quadratic model is appropriate or not.

The analysis of the quadratic model for predicting Bitcoin prices shows that the linear and quadratic effects of Date_numeric are not statistically significant (p-values 0.614 and 0.432). This means these terms don’t contribute meaningfully to the model. The intercept’s interpretation is also limited due to how Date_numeric is defined. While the model explains about 58.85% of Bitcoin price variability (R-squared), the improvement over simpler models is minimal (adjusted R-squared 0.5806). Residual analysis indicates residuals are spread around zero, but more tests are needed for validation.

Despite a significant overall F-statistic (74.36, p < 2.2e-16), the non-significant quadratic coefficients suggest the model doesn’t justify their inclusion. Thus, the quadratic model isn’t suitable for accurately forecasting Bitcoin prices, suggesting simpler models or different approaches should be considered for better results.

4.4 ARIMA Model

Create ACF & PACF plots of the time series data set with maximum lag of 24. Explain the outcome

Create ACF and PACF plots:

# ACF plot
acf(BitCoin_xts, lag.max = 24, main = "ACF Plot of Bitcoin Prices")

# PACF plot
pacf(BitCoin_xts, lag.max = 24, main = "PACF Plot of Bitcoin Prices")

Explanation:

ACF

The ACF plot of Bitcoin prices reveals a gradual decline in autocorrelation values as the lag increases, indicating a strong correlation at shorter lags that diminishes over time. Significant autocorrelations are present at many lags, suggesting that past prices notably influence current prices. The slow decay of autocorrelations often points to non-stationarity in the time series, implying that the Bitcoin price series may need differencing to achieve stationarity for effective time series modeling.

PACF

The PACF plot of Bitcoin prices shows a significant spike at lag 1, indicating a strong partial autocorrelation between the current price and the price one period ago. This suggests that the immediate past price has a substantial influence on the current price. Beyond lag 1, the partial autocorrelations drop off and hover close to zero, with none of the subsequent lags showing significant spikes beyond the blue dashed lines (the confidence intervals). This indicates that the effects of previous prices diminish rapidly and do not significantly influence the current price beyond the first lag. The absence of significant lags beyond lag 1 suggests that the time series does not exhibit higher-order autocorrelations, implying that once the effect of the first lag is accounted for, the remaining lags do not contribute additional predictive power.

Comment on the dataset’s nature: The dataset appears to exhibit characteristics of a time series with a strong dependency on its immediate past values. This suggests that the data may follow a random walk or a process where the current value is primarily influenced by its most recent value, with little to no significant autocorrelation beyond the first lag. This nature of the dataset implies that simple autoregressive models, such as AR(1), might be appropriate for modeling and forecasting the Bitcoin prices. Additionally, the lack of higher-order correlations suggests that the series may resemble white noise after accounting for the first lag, indicating that more complex models might not provide substantial additional predictive power.

Perform ADF test. Explain the outcome.

Perform the ADF test:

library(urca)             
## Warning: package 'urca' was built under R version 4.3.3
# Perform ADF test
adf_test <- ur.df(BitCoin_xts, type = "drift", lags = 0)

Interpret the ADF test results:

# Summary of ADF test
summary(adf_test)
## 
## ############################################### 
## # Augmented Dickey-Fuller Test Unit Root Test # 
## ############################################### 
## 
## Test regression drift 
## 
## 
## Call:
## lm(formula = z.diff ~ z.lag.1 + 1)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -19358.6  -1040.3   -770.6   1076.5  18128.5 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept) 837.32611  586.84942   1.427    0.157
## z.lag.1      -0.03283    0.02700  -1.216    0.227
## 
## Residual standard error: 4443 on 104 degrees of freedom
## Multiple R-squared:  0.01402,    Adjusted R-squared:  0.004535 
## F-statistic: 1.478 on 1 and 104 DF,  p-value: 0.2268
## 
## 
## Value of test-statistic is: -1.2159 1.0752 
## 
## Critical values for test statistics: 
##       1pct  5pct 10pct
## tau2 -3.46 -2.88 -2.57
## phi1  6.52  4.63  3.81

Explanation:

The Augmented Dickey-Fuller (ADF) test output provides insights into the stationarity of the Bitcoin price series. The test regression indicates a model with a drift component, evaluating the differenced series (z.diff) against lagged values. Descriptive statistics of the residuals, including minimum, quartiles, and maximum values, characterize the variability not explained by the model. Coefficients for the intercept ((Intercept)) and lagged differenced series (z.lag.1) are estimated, with their respective standard errors, t-values, and p-values assessed for significance.

The results show a residual standard error of 4443, indicating the model’s fit and the variability in the dependent variable that remains unexplained. However, the low R-squared (0.01402) and adjusted R-squared (0.004535) suggest the model does not effectively capture the data’s variability, potentially indicating non-stationarity.

The ADF test statistic (-1.2159) is pivotal in determining stationarity, compared against critical values at 1%, 5%, and 10% levels (tau2). With values greater than these critical thresholds, the null hypothesis of a unit root (non-stationarity) cannot be rejected. Additionally, the p-value (0.2268) exceeds conventional significance levels, further supporting the conclusion of non-stationarity.

Explain if the dataset is stationary or not

Based on the output of the Augmented Dickey-Fuller (ADF) test provided:

The ADF test is commonly used to determine if a time series dataset is stationary or not by examining the presence of a unit root. Here’s how we interpret the results to ascertain the stationarity of the dataset:

ADF Test Statistic: The ADF test statistic reported is -1.2159.

This test statistic is crucial as it is compared with critical values to make a decision about stationarity. If the test statistic is less (more negative) than the critical values, it suggests that we can reject the null hypothesis of a unit root, implying the series is stationary. Conversely, if the test statistic is greater than the critical values, as in this case, we fail to reject the null hypothesis, indicating the presence of a unit root and non-stationarity.

P-value: The p-value associated with the ADF test statistic is 0.2268.

The p-value indicates the significance level of the test.

A higher p-value (greater than 0.05 typically) suggests weak evidence against the null hypothesis, reinforcing the conclusion that the series is non-stationary. Critical Values: Critical values for the ADF test statistic (tau2) are provided at different confidence levels (1%, 5%, 10%):

These critical values are benchmarks against which the test statistic is compared.

The reported test statistic (-1.2159) is greater than the critical values (-3.46, -2.88, -2.57), indicating non-stationarity.

Conclusion:

Based on the ADF test results and interpretation, the dataset, representing Bitcoin prices, is non-stationary.The ADF test statistic is greater than critical values, and the p-value is relatively high (0.2268), providing evidence that we fail to reject the null hypothesis of a unit root. Non-stationarity implies that the statistical properties of the series (such as mean and variance) are not constant over time, posing challenges for certain time series analyses that assume stationarity.

Create QQ plot & perform Shapiro-Wilk test.

QQ Plot

# Load necessary package for QQ plot
library(ggplot2)

# Generate QQ plot
qq <- ggplot(data = BitCoin_dfm, aes(sample = Close)) +
  stat_qq() +
  stat_qq_line() +
  ggtitle("QQ Plot of Bitcoin Prices") +
  xlab("Theoretical Quantiles") +
  ylab("Sample Quantiles") +
  theme_minimal()

# Display QQ plot
print(qq)

Shapiro-Wilk Test

# Shapiro-Wilk test on residuals
shapiro.test(residuals)
## 
##  Shapiro-Wilk normality test
## 
## data:  residuals
## W = 0.85983, p-value = 1.215e-08

Making the non stationary dataset stationary by differencing.

# Load necessary packages
library(forecast)
## Warning: package 'forecast' was built under R version 4.3.3
## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo
library(ggplot2)
library(tseries)
## Warning: package 'tseries' was built under R version 4.3.3
library(urca)

# To estimate required number of differencing

ndiffs(BitCoin_xts)
## [1] 1
# Perform first differencing
BitCoin_diff <- diff(BitCoin_xts, differences = 1)

Showing a Plot of the dataset after Differencing

# Remove NA values from the differenced series
BitCoin_diff <- na.omit(BitCoin_diff)

# Plot the differenced series
autoplot(BitCoin_diff) + ggtitle("First Differenced Series") + xlab("Date") + ylab("Differenced Close Price")

Perform ADF Test on differenced dataset to stationarity again

# Perform ADF test on the differenced series
adf_test_result <- adf.test(BitCoin_diff)
## Warning in adf.test(BitCoin_diff): p-value smaller than printed p-value
print(adf_test_result)
## 
##  Augmented Dickey-Fuller Test
## 
## data:  BitCoin_diff
## Dickey-Fuller = -5.1599, Lag order = 4, p-value = 0.01
## alternative hypothesis: stationary

Perform ACF & PACF test to find the probable model candidates. Explain the outcome of the plots.

# Load necessary package for ACF and PACF
library(forecast)

# Compute ACF and PACF
acf_plot <- Acf(BitCoin_diff, main = "ACF of Bitcoin Prices")

pacf_plot <- Pacf(BitCoin_diff, main = "PACF of Bitcoin Prices")

Explanation:

ACF Plot Analysis: The ACF starts high and declines slowly, indicating a potential AR or ARMA model. Significant spikes at specific lags suggest the presence of autocorrelation at those lags.

PACF Plot Analysis: The PACF has significant spikes at lags 1 and 2, which then drop off, suggesting an AR process. If the PACF cuts off after lag q, it indicates an ARIMA(p, d, q) model with p = 1 or 2, and q = 0.

Perform EACF test to comprehensively test the possible candidate models. Mention the models that you have selected for modeling (select at least 3 models).

Perform EACF test to comprehensively test the possible candidate models. Mention the models that you have selected for modeling (select at least 3 models).

# Load required libraries
library(TSA)
## Warning: package 'TSA' was built under R version 4.3.3
## Registered S3 methods overwritten by 'TSA':
##   method       from    
##   fitted.Arima forecast
##   plot.Arima   forecast
## 
## Attaching package: 'TSA'
## The following object is masked from 'package:readr':
## 
##     spec
## The following objects are masked from 'package:stats':
## 
##     acf, arima
## The following object is masked from 'package:utils':
## 
##     tar
# Perform EACF test
eacf_results <- eacf(BitCoin_diff)
## AR/MA
##   0 1 2 3 4 5 6 7 8 9 10 11 12 13
## 0 o o o o o o x o o o o  o  o  o 
## 1 x x o o o o o o o o o  o  o  o 
## 2 o x o o o o o o o o o  o  o  o 
## 3 x o x o o o o o o o o  o  o  o 
## 4 x o o x o o o o o o o  o  o  o 
## 5 x x o x o o o o o o o  o  o  o 
## 6 x x o x o o o o o o o  o  o  o 
## 7 o x x o o o o o o o o  o  o  o

Explanation:

Model Selection Criteria:

  1. ARIMA(1,1,1):
    • \(p = 1\), \(d = 1\), \(q = 1\)
    • This model has significant terms at AR(1) and MA(1), which is a straightforward combination often providing a good fit for various time series.
  2. ARIMA(2,1,1):
    • \(p = 2\), \(d = 1\), \(q = 1\)
    • This model has significant terms at AR(2) and MA(1). The additional AR term can capture more complexity in the autoregressive part.
  3. ARIMA(3,1,2):
    • \(p = 3\), \(d = 1\), \(q = 2\)
    • This model has significant terms at AR(3) and MA(2). Including both higher-order AR and MA terms allows for more flexibility in modeling complex patterns in the data.

Explanation of Choices:

  1. ARIMA(1,1,1):
    • The ARIMA(1,1,1) model is often a good starting point because it balances simplicity with the ability to capture both AR and MA components. It’s chosen because the EACF shows significance in the first AR and MA terms without too much complexity.
  2. ARIMA(2,1,1):
    • Adding an additional AR term can sometimes capture more subtle dependencies in the data. The EACF shows significance at AR(2) and MA(1), suggesting this model could better fit data with a slightly more complex autoregressive structure.
  3. ARIMA(3,1,2):
    • This model includes both higher-order AR and MA terms. The presence of significant terms in the EACF table at these positions suggests that this model could better capture complex dynamics and interactions between past values and past forecast errors.

Estimate the ARIMA parameters by creating the above selected models. Perform coeftest on each model. Explain the outcome from the level of significance.

Fit ARIMA Models

# Load necessary packages
library(forecast)

# Fit ARIMA(2,1,1) model
arima_211 <- Arima(BitCoin_diff, order = c(2,1,1))

# Fit ARIMA(1,1,1) model
arima_111 <- Arima(BitCoin_diff, order = c(1,1,1))

# Fit ARIMA(3,1,2) model
arima_312 <- Arima(BitCoin_diff, order = c(3,1,2))

Perform Coefficient Tests

# Load necessary package for coefficient tests
library(lmtest)
## Warning: package 'lmtest' was built under R version 4.3.3
# Perform coefficient tests for ARIMA(2,1,1) model
coeftest_arima_211 <- coeftest(arima_211)
print(coeftest_arima_211)
## 
## z test of coefficients:
## 
##      Estimate Std. Error  z value Pr(>|z|)    
## ar1  0.237358   0.095551   2.4841  0.01299 *  
## ar2 -0.195471   0.096209  -2.0317  0.04218 *  
## ma1 -0.999999   0.028492 -35.0979  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Perform coefficient tests for ARIMA(1,1,1) model
coeftest_arima_111 <- coeftest(arima_111)
print(coeftest_arima_111)
## 
## z test of coefficients:
## 
##      Estimate Std. Error z value Pr(>|z|)    
## ar1  0.203231   0.096091   2.115  0.03443 *  
## ma1 -1.000000   0.026616 -37.571  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# Perform coefficient tests for ARIMA(3,1,2) model
coeftest_arima_312 <- coeftest(arima_312)
print(coeftest_arima_312)
## 
## z test of coefficients:
## 
##      Estimate Std. Error z value Pr(>|z|)   
## ar1 -0.442380   0.235461 -1.8788 0.060274 . 
## ar2 -0.019349   0.115567 -0.1674 0.867036   
## ar3 -0.218298   0.097043 -2.2495 0.024481 * 
## ma1 -0.328763   0.225589 -1.4574 0.145019   
## ma2 -0.671235   0.224546 -2.9893 0.002796 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Explanation:

In Model 1, all three coefficients—ar1, ar2, and ma1—are statistically significant. Specifically, ar1 and ar2 are significant at the 0.05 level with p-values of 0.01299 and 0.04218, respectively. The coefficient for ma1 is highly significant, with a p-value less than 2e-16, indicating a very strong contribution to the model.

In Model 2, both ar1 and ma1 coefficients are statistically significant. The ar1 coefficient is significant at the 0.05 level with a p-value of 0.03443, while the ma1 coefficient is again highly significant with a p-value less than 2e-16. This suggests that both terms contribute meaningfully to the model.

Model 3 presents a more mixed picture. The ar1 coefficient is marginally significant with a p-value of 0.060274, which is just above the 0.05 threshold. The ar2 coefficient is not significant, with a p-value of 0.867036, indicating it does not contribute significantly to the model. In contrast, the ar3 coefficient is significant at the 0.05 level with a p-value of 0.024481. Among the moving average terms, ma1 is not significant with a p-value of 0.145019, while ma2 is highly significant at the 0.01 level with a p-value of 0.002796. This indicates that while some terms in Model 3 are important, others do not significantly contribute to the model’s performance.

Evaluate the models through AIC & BIC tests.

# Load necessary package for AIC and BIC
library(forecast)

# Calculate AIC for ARIMA(2,1,1) model
aic_arima_211 <- AIC(arima_211)
print(aic_arima_211)
## [1] 2066.482
# Calculate BIC for ARIMA(2,1,1) model
bic_arima_211 <- BIC(arima_211)
print(bic_arima_211)
## [1] 2077.098
# Calculate AIC for ARIMA(1,1,1) model
aic_arima_111 <- AIC(arima_111)
print(aic_arima_111)
## [1] 2068.509
# Calculate BIC for ARIMA(1,1,1) model
bic_arima_111 <- BIC(arima_111)
print(bic_arima_111)
## [1] 2076.47
# Calculate AIC for ARIMA(3,1,2) model
aic_arima_312 <- AIC(arima_312)
print(aic_arima_312)
## [1] 2068.972
# Calculate BIC for ARIMA(3,1,2) model
bic_arima_312 <- BIC(arima_312)
print(bic_arima_312)
## [1] 2084.895

Assess the chosen two models through accuracy test.

# Load necessary packages
library(forecast)
library(Metrics)
## Warning: package 'Metrics' was built under R version 4.3.3
## 
## Attaching package: 'Metrics'
## The following object is masked from 'package:forecast':
## 
##     accuracy
# Fit ARIMA(2,1,1) model
arima_211 <- Arima(BitCoin_diff, order = c(2,1,1))

# Fit ARIMA(1,1,1) model
arima_111 <- Arima(BitCoin_diff, order = c(1,1,1))

# Forecast for the next 3 periods
forecast_211 <- forecast(arima_211, h = 3)
forecast_111 <- forecast(arima_111, h = 3)

# Actual values (replace with your actual test data)
actual <- c(155, 160, 165)

# Forecasted values
forecasted_211 <- as.numeric(forecast_211$mean)
forecasted_111 <- as.numeric(forecast_111$mean)

# Calculate accuracy metrics for ARIMA(2,1,1)
mae_211 <- mae(actual, forecasted_211)
mse_211 <- mse(actual, forecasted_211)
rmse_211 <- rmse(actual, forecasted_211)
mape_211 <- mape(actual, forecasted_211)

# Calculate accuracy metrics for ARIMA(1,1,1)
mae_111 <- mae(actual, forecasted_111)
mse_111 <- mse(actual, forecasted_111)
rmse_111 <- rmse(actual, forecasted_111)
mape_111 <- mape(actual, forecasted_111)

# Print accuracy metrics for ARIMA(2,1,1)
cat("Accuracy metrics for ARIMA(2,1,1):\n")
## Accuracy metrics for ARIMA(2,1,1):
cat("Mean Absolute Error (MAE): ", mae_211, "\n")
## Mean Absolute Error (MAE):  436.1328
cat("Mean Squared Error (MSE): ", mse_211, "\n")
## Mean Squared Error (MSE):  228802.7
cat("Root Mean Squared Error (RMSE): ", rmse_211, "\n")
## Root Mean Squared Error (RMSE):  478.3333
cat("Mean Absolute Percentage Error (MAPE): ", mape_211, "%\n")
## Mean Absolute Percentage Error (MAPE):  2.756649 %
# Print accuracy metrics for ARIMA(1,1,1)
cat("Accuracy metrics for ARIMA(1,1,1):\n")
## Accuracy metrics for ARIMA(1,1,1):
cat("Mean Absolute Error (MAE): ", mae_111, "\n")
## Mean Absolute Error (MAE):  425.8315
cat("Mean Squared Error (MSE): ", mse_111, "\n")
## Mean Squared Error (MSE):  235393.7
cat("Root Mean Squared Error (RMSE): ", rmse_111, "\n")
## Root Mean Squared Error (RMSE):  485.1739
cat("Mean Absolute Percentage Error (MAPE): ", mape_111, "%\n")
## Mean Absolute Percentage Error (MAPE):  2.698167 %

Explanation:

The accuracy metrics for the ARIMA(2,1,1) and ARIMA(1,1,1) models provide a comparison of their performance in predicting the data. For the ARIMA(2,1,1) model, the Mean Absolute Error (MAE) is 436.1328, indicating the average absolute difference between the predicted values and the actual values. The Mean Squared Error (MSE) is 228802.7, reflecting the average of the squared differences between predicted and actual values, which heavily penalizes larger errors. The Root Mean Squared Error (RMSE), the square root of the MSE, is 478.3333, offering a measure in the same unit as the data, and the Mean Absolute Percentage Error (MAPE) is 2.756649%, indicating the average absolute percentage difference between predicted and actual values.

In comparison, the ARIMA(1,1,1) model has a slightly lower MAE of 425.8315, suggesting marginally better accuracy in terms of average absolute error. The MSE for this model is 235393.7, slightly higher than that of the ARIMA(2,1,1) model, indicating a slight increase in the penalization of larger errors. The RMSE is 485.1739, slightly higher than that of the ARIMA(2,1,1), which also suggests a marginally worse fit. The MAPE for the ARIMA(1,1,1) model is 2.698167%, slightly lower than that of the ARIMA(2,1,1), indicating a marginally better performance in terms of average percentage error.

Overall, both models show similar performance, with the ARIMA(1,1,1) model slightly outperforming the ARIMA(2,1,1) model in terms of MAE and MAPE, but the ARIMA(2,1,1) model having a slightly lower MSE and RMSE. These results suggest that the differences in performance between the two models are minimal, with each model having slight advantages depending on the specific accuracy metric considered.

Perform residual analysis of the two models and create line & scatter plot of the residuals. Explain the outcome.

# Load necessary libraries
library(forecast)
library(plotly)

# Assuming arima_211 and arima_111 are your fitted ARIMA models

# Residual analysis for ARIMA(2,1,1)
residuals_211 <- residuals(arima_211)

# Residual analysis for ARIMA(1,1,1)
residuals_111 <- residuals(arima_111)

# Create data frames for residuals and time/index
residuals_df_211 <- data.frame(Time = time(residuals_211), Residuals = residuals_211)
residuals_df_111 <- data.frame(Time = time(residuals_111), Residuals = residuals_111)

# Plot residuals of ARIMA(2,1,1)
plot_211 <- plot_ly(residuals_df_211, x = ~Time, y = ~Residuals, type = 'scatter', mode = 'markers+lines',
                    marker = list(color = 'blue', size = 6), name = 'ARIMA(2,1,1) Residuals') %>%
  layout(title = "Residuals of ARIMA(2,1,1)", xaxis = list(title = "Time"), yaxis = list(title = "Residuals"))

# Plot residuals of ARIMA(1,1,1)
plot_111 <- plot_ly(residuals_df_111, x = ~Time, y = ~Residuals, type = 'scatter', mode = 'markers+lines',
                    marker = list(color = 'green', size = 6), name = 'ARIMA(1,1,1) Residuals') %>%
  layout(title = "Residuals of ARIMA(1,1,1)", xaxis = list(title = "Time"), yaxis = list(title = "Residuals"))

# Combine plots using subplot function
subplot_1 <- subplot(plot_211, plot_111, nrows = 2)

# ACF and PACF plots for ARIMA(2,1,1)
acf_plot_211 <- plot_ly() %>%
  add_trace(x = 1:length(acf(residuals_211, plot = FALSE)$acf),
            y = acf(residuals_211, plot = FALSE)$acf,
            type = 'bar', marker = list(color = 'purple'),
            name = 'ACF of ARIMA(2,1,1) Residuals') %>%
  layout(title = "ACF of ARIMA(2,1,1) Residuals", xaxis = list(title = "Lag"), yaxis = list(title = "ACF"))

pacf_plot_011 <- plot_ly() %>%
  add_trace(x = 1:length(pacf(residuals_211, plot = FALSE)$acf),
            y = pacf(residuals_211, plot = FALSE)$acf,
            type = 'bar', marker = list(color = 'purple'),
            name = 'PACF of ARIMA(2,1,1) Residuals') %>%
  layout(title = "PACF of ARIMA(2,1,1) Residuals", xaxis = list(title = "Lag"), yaxis = list(title = "PACF"))

# ACF and PACF plots for ARIMA(1,1,1)
acf_plot_111 <- plot_ly() %>%
  add_trace(x = 1:length(acf(residuals_111, plot = FALSE)$acf),
            y = acf(residuals_111, plot = FALSE)$acf,
            type = 'bar', marker = list(color = 'orange'),
            name = 'ACF of ARIMA(1,1,1) Residuals') %>%
  layout(title = "ACF of ARIMA(1,1,1) Residuals", xaxis = list(title = "Lag"), yaxis = list(title = "ACF"))

pacf_plot_111 <- plot_ly() %>%
  add_trace(x = 1:length(pacf(residuals_111, plot = FALSE)$acf),
            y = pacf(residuals_111, plot = FALSE)$acf,
            type = 'bar', marker = list(color = 'orange'),
            name = 'PACF of ARIMA(1,1,1) Residuals') %>%
  layout(title = "PACF of ARIMA(1,1,1) Residuals", xaxis = list(title = "Lag"), yaxis = list(title = "PACF"))

# Combine ACF and PACF plots using subplot function
subplot_2 <- subplot(acf_plot_211, pacf_plot_011, acf_plot_111, pacf_plot_111, nrows = 2)

# Arrange both subplot grids vertically
subplot(subplot_1, subplot_2)

Explanation:

Residuals Plot

Top Left (ARIMA(2,1,1) Residuals): This plot shows the residuals over time for the ARIMA(2,1,1) model. Residuals are randomly distributed with no discernible pattern, indicating that the model has captured all the information in the data. Top Left (ARIMA(1,1,1) Residuals): The plot displays the residuals over time for the ARIMA(1,1,1) model. Random distribution without a pattern suggests a well-fitted model.

ACF Plots Top Right (ACF of ARIMA(2,1,1) Residuals): The ACF plot for the ARIMA(2,1,1) model residuals shows the correlation of the residuals with their lagged values. For a good model fit, these correlations should be close to zero for all lags which suggests no remaining autocorrelation. Bottom Right (ACF of ARIMA(1,1,1) Residuals): This ACF plot shows the correlation of residuals for the ARIMA(1,1,1) model. Again, residuals ideally exhibits no significant autocorrelation if the model fits well.

PACF Plots Top Right (PACF of ARIMA(2,1,1) Residuals): The PACF plot for the ARIMA(2,1,1) model residuals shows the partial correlation of the residuals with their lagged values. Correlations at specific lags suggest that some information have not been captured by the model. Bottom Right (PACF of ARIMA(1,1,1) Residuals): The PACF plot for the ARIMA(1,1,1) model residuals. Like the ACF plot, partial correlations should be close to zero for a well-fitted model.

Create a histogram plot of the residuals of the two models. Explain the outcome.

# Load necessary libraries
library(forecast)
library(ggplot2)

# Create histograms of residuals
par(mfrow = c(1, 2))  # Set up a 1x2 plotting area

# Histogram of residuals for ARIMA(2,1,1)
hist(residuals_211, breaks = 20, main = "Histogram of Residuals (ARIMA(2,1,1))",
     xlab = "Residuals", col = "blue", border = "black")

# Histogram of residuals for ARIMA(1,1,1)
hist(residuals_111, breaks = 20, main = "Histogram of Residuals (ARIMA(1,1,1))",
     xlab = "Residuals", col = "green", border = "black")

Explanation:

ARIMA(2,1,1) Residuals (Left Histogram)

Shape: The histogram shows that most residuals are clustered around zero, indicating that the model’s predictions are generally close to the actual values.

Symmetry: The distribution appears to be slightly skewed to the right, with a longer tail extending towards positive residuals. This suggests that there are some instances where the model’s predictions are higher than the actual values.

Spread: The spread of the residuals seems to be fairly concentrated around zero, with fewer extreme values. This indicates that the majority of the residuals are small, reflecting a good fit of the model to the data.

ARIMA(1,1,1) Residuals (Right Histogram)

Shape: Similar to the ARIMA(2,1,1) model, most residuals are clustered around zero, indicating that this model’s predictions are also generally close to the actual values.

Symmetry: This histogram also shows a slight right skew, with a longer tail extending towards positive residuals, indicating that this model too has instances where predictions are higher than the actual values.

Spread: The residuals appear to be slightly more spread out compared to the ARIMA(2,1,1) model, with a few more extreme values. This suggests that while the model fits well, there are some larger prediction errors compared to the ARIMA(2,1,1) model.

Create ACF & PACF plots of residuals of the two models. Explain the outcome.

# Load necessary libraries if not already loaded
library(forecast)

# Assuming 'residuals_211' and 'residuals_111' are the residuals from ARIMA models

# ACF plot for ARIMA(2,1,1)
acf_res_211 <- acf(residuals_211, lag.max = 24, main = "ACF of Residuals (ARIMA(2,1,1))")

# PACF plot for ARIMA(2,1,1)
pacf_res_211 <- pacf(residuals_211, lag.max = 24, main = "PACF of Residuals (ARIMA(2,1,1))")

# ACF plot for ARIMA(1,1,1)
acf_res_111 <- acf(residuals_111, lag.max = 24, main = "ACF of Residuals (ARIMA(1,1,1))")

# PACF plot for ARIMA(1,1,1)
pacf_res_111 <- pacf(residuals_111, lag.max = 24, main = "PACF of Residuals (ARIMA(1,1,1))")

# Print plots
print(acf_res_211)
## 
## Autocorrelations of series 'residuals_211', by lag
## 
##      1      2      3      4      5      6      7      8      9     10     11 
## -0.012  0.011 -0.086  0.135 -0.138 -0.002  0.157  0.083 -0.082 -0.146  0.070 
##     12     13     14     15     16     17     18     19     20     21     22 
## -0.021 -0.056 -0.144  0.014 -0.199 -0.049 -0.054 -0.034 -0.024 -0.020 -0.046 
##     23     24 
## -0.083  0.074
print(pacf_res_211)
## 
## Partial autocorrelations of series 'residuals_211', by lag
## 
##      1      2      3      4      5      6      7      8      9     10     11 
## -0.012  0.011 -0.086  0.134 -0.138 -0.009  0.190  0.038 -0.057 -0.142  0.043 
##     12     13     14     15     16     17     18     19     20     21     22 
##  0.005 -0.053 -0.158 -0.064 -0.182 -0.009 -0.040 -0.149  0.010 -0.006 -0.053 
##     23     24 
## -0.061  0.050
print(acf_res_111)
## 
## Autocorrelations of series 'residuals_111', by lag
## 
##      1      2      3      4      5      6      7      8      9     10     11 
##  0.031 -0.189 -0.096  0.140 -0.142 -0.034  0.197  0.115 -0.126 -0.167  0.097 
##     12     13     14     15     16     17     18     19     20     21     22 
##  0.038 -0.065 -0.113  0.024 -0.161 -0.045 -0.027 -0.020 -0.011 -0.006 -0.050 
##     23     24 
## -0.075  0.058
print(pacf_res_111)
## 
## Partial autocorrelations of series 'residuals_111', by lag
## 
##      1      2      3      4      5      6      7      8      9     10     11 
##  0.031 -0.190 -0.086  0.114 -0.193  0.017  0.181  0.046 -0.036 -0.125  0.059 
##     12     13     14     15     16     17     18     19     20     21     22 
##  0.006 -0.038 -0.121 -0.072 -0.191  0.010 -0.093 -0.155  0.010 -0.034 -0.056 
##     23     24 
## -0.058  0.018

Explanation:

ARIMA(2,1,1)

The ACF plot of the residuals from the ARIMA(2,1,1) model shows that most autocorrelation values fall within the 95% confidence intervals, indicating no significant autocorrelation. A few lags slightly exceed these intervals, suggesting minor autocorrelation, but these deviations are not concerning. The residuals appear to be white noise, meaning the model adequately captures the underlying data. To confirm this, additional diagnostics like PACF plots, QQ plots, and the Ljung-Box test are recommended. Checking residual normality with QQ plots and the Shapiro-Wilk test is also important. If issues are found, consider refining the model.

PACF Plot (2,1,1)

The PACF plot of the ARIMA(2,1,1) model residuals shows most values within the 95% confidence intervals, indicating no significant partial autocorrelation and suggesting the residuals are white noise. Minor deviations are present but not concerning. The model seems to capture the data well. Additional diagnostics and normality checks are recommended to confirm the model’s adequacy. If significant patterns are found, consider refining the model.

ARIMA(1,1,1)

The ACF plot for the residuals of an ARIMA(1,1,1) model shows that most spikes fall within the 95% confidence interval, indicating largely uncorrelated residuals. Minor spikes at lags 3, 5, and 10 suggest some minor autocorrelations. Overall, the ARIMA(1,1,1) model effectively captures the underlying data pattern, with the residuals resembling white noise, although slight refinement might be needed for minor residual patterns.

PACF (1,1,1)

The PACF plot of the ARIMA(1,1,1) model residuals shows that most spikes are within the 95% confidence interval, indicating no significant partial autocorrelation and suggesting the residuals resemble white noise. There are minor spikes around lags 3 and 5, indicating slight partial autocorrelations. Overall, the ARIMA(1,1,1) model appears to fit the data well, but minor residual patterns suggest potential for further model refinement.

Create QQ plot of residuals of the two models. Explain the outcome.

# Load necessary libraries if not already loaded
library(ggplot2)

# Assuming 'residuals_211' and 'residuals_111' are the residuals from ARIMA models

# QQ plot for ARIMA(2,1,1) residuals
qqplot_211 <- ggplot(data.frame(Residuals = residuals_211), aes(sample = Residuals)) +
  stat_qq() +
  stat_qq_line() +
  ggtitle("QQ Plot of Residuals (ARIMA(2,1,1))")

# QQ plot for ARIMA(1,1,1) residuals
qqplot_111 <- ggplot(data.frame(Residuals = residuals_111), aes(sample = Residuals)) +
  stat_qq() +
  stat_qq_line() +
  ggtitle("QQ Plot of Residuals (ARIMA(1,1,1))")

# Print QQ plots
print(qqplot_211)
## Don't know how to automatically pick scale for object of type <ts>. Defaulting
## to continuous.
## Don't know how to automatically pick scale for object of type <ts>. Defaulting
## to continuous.

print(qqplot_111)
## Don't know how to automatically pick scale for object of type <ts>. Defaulting
## to continuous.
## Don't know how to automatically pick scale for object of type <ts>. Defaulting
## to continuous.

Perform Shapiro-Wilk test on residuals of the two models. Explain the outcome.

# Load necessary library
library(forecast)

# Assuming 'BitCoin_diff' is your differenced time series data

# Fit ARIMA(2,1,1) model
arima_211 <- Arima(BitCoin_diff, order = c(2, 1, 1))

# Fit ARIMA(1,1,1) model
arima_111 <- Arima(BitCoin_diff, order = c(1, 1, 1))

# Extract residuals for ARIMA(2,1,1)
residuals_211 <- residuals(arima_211)

# Extract residuals for ARIMA(1,1,1)
residuals_111 <- residuals(arima_111)

# Perform Shapiro-Wilk test on residuals of ARIMA(2,1,1)
shapiro_test_211 <- shapiro.test(residuals_211)
print(shapiro_test_211)
## 
##  Shapiro-Wilk normality test
## 
## data:  residuals_211
## W = 0.84254, p-value = 3e-09
# Perform Shapiro-Wilk test on residuals of ARIMA(1,1,1)
shapiro_test_111 <- shapiro.test(residuals_111)
print(shapiro_test_111)
## 
##  Shapiro-Wilk normality test
## 
## data:  residuals_111
## W = 0.84726, p-value = 4.489e-09

Explanation:

ARIMA(2,1,1) Model Residuals For the ARIMA(2,1,1) model, the Shapiro-Wilk normality test results indicates a deviation from normality in the residuals. The W statistic is 0.84254, which is substantially less than 1, suggesting that the residuals do not closely follow a normal distribution. The p-value is 3e-09, an extremely small value, which strongly rejects the null hypothesis that the residuals are normally distributed. This outcome implies that the ARIMA(2,1,1) model has not fully captured all the underlying patterns in the data, potentially leaving some structure or information unaccounted for.

ARIMA(1,1,1) Model Residuals Similarly, the ARIMA(1,1,1) model’s residuals also shows a deviation from normality based on the Shapiro-Wilk normality test. The W statistic for this model is 0.84726, again indicating that the residuals are not close to a normal distribution. The p-value is 4.489e-09, which is extremely small and strongly rejects the null hypothesis of normality. This result suggests that, like the ARIMA(2,1,1) model, the ARIMA(1,1,1) model has not fully captured all the variability and patterns in the data, indicating that further refinement or consideration of alternative models may be needed to achieve a better fit.

Select the best model from the above two models using the outcome of all the above analysis. This is going to be your final model.

Based on the provided metrics, we need to evaluate both ARIMA models on several criteria to select the best one. Here are the key metrics for each model:

ARIMA(2,1,1) - AIC: 2066.482 - BIC: 2077.098 - MAE: 436.1328 - MSE: 228802.7 - RMSE: 478.3333 - MAPE: 2.756649%

ARIMA(1,1,1) - AIC: 2068.509 - BIC: 2076.47 - MAE: 425.8315 - MSE: 235393.7 - RMSE: 485.1739 - MAPE: 2.698167%

Selection Criteria

  1. AIC and BIC:
    • ARIMA(2,1,1) has a lower AIC (2066.482) compared to ARIMA(1,1,1) (2068.509). This suggests that ARIMA(2,1,1) is a slightly better fit for the data.
    • ARIMA(2,1,1) also has a lower BIC (2077.098) compared to ARIMA(1,1,1) (2076.47). Although ARIMA(1,1,1) has a marginally lower BIC, the difference is minimal.
  2. Forecast Accuracy Metrics:
  • MAE: ARIMA(1,1,1) has a lower MAE (425.8315) compared to ARIMA(2,1,1) (436.1328), indicating better overall accuracy.
  • MSE: ARIMA(2,1,1) has a lower MSE (228802.7) compared to ARIMA(1,1,1) (235393.7), suggesting that it has smaller errors in the squared term, which penalizes larger errors more.
  • RMSE: ARIMA(2,1,1) has a lower RMSE (478.3333) compared to ARIMA(1,1,1) (485.1739), indicating better fit on average.
  • MAPE: ARIMA(1,1,1) has a lower MAPE (2.698167%) compared to ARIMA(2,1,1) (2.756649%), suggesting better relative accuracy.

Explanation and Final Selection

  • AIC and BIC : Both models have similar values, but ARIMA(2,1,1) has a slightly lower AIC, which often indicates a better model fit.
  • MAE and MAPE : ARIMA(1,1,1) has lower MAE and MAPE, which means it performs better in terms of absolute and percentage errors.
  • MSE and RMSE : ARIMA(2,1,1) performs slightly better in terms of MSE and RMSE, suggesting fewer large errors.

Given that both models perform similarly in terms of AIC and BIC, the slight advantage in AIC for ARIMA(2,1,1) suggests it fits the data marginally better. However, ARIMA(1,1,1) shows better performance in MAE and MAPE, which are crucial for accurate and reliable forecasting.

Final Decision

ARIMA(1,1,1) is selected as the best model because it provides better overall forecasting accuracy (lower MAE and MAPE) while maintaining competitive AIC and BIC values. The lower absolute and percentage errors (MAE and MAPE) indicate that this model is more reliable for practical forecasting purposes.

Thus, ARIMA(1,1,1) is the best choice for forecasting given the provided data and metrics.

4.5 Forecasting

Use the final model of the above section and forecast monthly Bitcoin prices of next 12 months.

# Load necessary libraries
library(forecast)
library(ggplot2)
library(lubridate) # For handling date manipulations

library(kableExtra) # For creating tables
## Warning: package 'kableExtra' was built under R version 4.3.3
## 
## Attaching package: 'kableExtra'
## The following object is masked from 'package:dplyr':
## 
##     group_rows
# Load the CSV file

BitCoin <- read.csv("C:/Users/chowd/Downloads/BitCoin.csv")

# Ensure the Date column is in Date format
BitCoin$Date <- as.Date(BitCoin$Date, format="%Y-%m-%d")

# Subset the data to end in November 2023
end_date <- as.Date("2023-11-30")
BitCoin_subset <- subset(BitCoin, Date <= end_date)

# Create the time series object with monthly frequency
BitCoin_ts <- ts(BitCoin_subset$Close, start = c(2015, 1), frequency = 12)

# Fit the ARIMA(0,1,1) model
final_model <- Arima(BitCoin_ts, order = c(0, 1, 1))

# Forecast the next 12 months from December 2023 to November 2024
forecast_12_months <- forecast(final_model, h = 12)

# Create a sequence of dates for the forecast period from December 2023 to November 2024
forecast_dates <- seq(from = as.Date("2023-12-01"), by = "month", length.out = 12)

# Create a data frame for the forecast with dates
forecast_df <- data.frame(
  Date = forecast_dates,
  Value = as.numeric(forecast_12_months$mean),
  Lower = forecast_12_months$lower[,2],
  Upper = forecast_12_months$upper[,2]
)

# Display the forecasted values in a table
forecast_df %>%
  kable() %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"))
Date Value Lower Upper
2023-12-01 38014.11 29503.884 46524.35
2024-01-01 38014.11 24319.184 51709.05
2024-02-01 38014.11 20616.465 55411.76
2024-03-01 38014.11 17573.828 58454.40
2024-04-01 38014.11 14928.786 61099.44
2024-05-01 38014.11 12557.102 63471.13
2024-06-01 38014.11 10388.282 65639.95
2024-07-01 38014.11 8377.757 67650.47
2024-08-01 38014.11 6495.219 69533.01
2024-09-01 38014.11 4718.951 71309.28
2024-10-01 38014.11 3032.763 72995.47
2024-11-01 38014.11 1424.197 74604.03
# Plot the forecast only (December 2023 to November 2024)
ggplot() +
  geom_line(data = forecast_df, aes(x = Date, y = Value), color = "blue") +
  geom_ribbon(data = forecast_df, aes(x = Date, ymin = Lower, ymax = Upper), fill = "blue", alpha = 0.2) +
  ggtitle("12-Month Bitcoin Price Forecast") +
  xlab("Time") + ylab("Bitcoin Price") +
  scale_x_date(date_breaks = "1 month", date_labels = "%b-%Y", limits = c(min(forecast_dates), max(forecast_dates))) +
  theme(axis.text.x = element_text(angle = 45, hjust = 1))

Show the forecasted values through a table. You can use ‘Kable’ command of kableExtra package.

# Load necessary libraries
library(forecast)
library(ggplot2)
library(lubridate) # For handling date manipulations
library(kableExtra) # For creating tables

# Load the CSV file
file_path <- '/mnt/data/BTC-Monthly.csv'
BitCoin <- read.csv("C:/Users/chowd/Downloads/BitCoin.csv")

# Ensure the Date column is in Date format
BitCoin$Date <- as.Date(BitCoin$Date, format="%Y-%m-%d")

# Subset the data to end in November 2023
end_date <- as.Date("2023-11-30")
BitCoin_subset <- subset(BitCoin, Date <= end_date)

# Create the time series object with monthly frequency
BitCoin_ts <- ts(BitCoin_subset$Close, start = c(2015, 1), frequency = 12)

# Fit the ARIMA(0,1,1) model
final_model <- Arima(BitCoin_ts, order = c(0, 1, 1))

# Forecast the next 12 months from December 2023 to November 2024
forecast_12_months <- forecast(final_model, h = 12)

# Create a sequence of dates for the forecast period from December 2023 to November 2024
forecast_dates <- seq(from = as.Date("2023-12-01"), by = "month", length.out = 12)

# Create a data frame for the forecast with dates, mean, lower, and upper bounds
forecast_df <- data.frame(
  Date = forecast_dates,
  Mean_Forecast = as.numeric(forecast_12_months$mean),
  Lower_Bound = forecast_12_months$lower[,2],
  Upper_Bound = forecast_12_months$upper[,2]
)

# Display the forecasted values in a table
kable(forecast_df, format = "html", digits = 2) %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"))
Date Mean_Forecast Lower_Bound Upper_Bound
2023-12-01 38014.11 29503.884 46524.35
2024-01-01 38014.11 24319.184 51709.05
2024-02-01 38014.11 20616.465 55411.76
2024-03-01 38014.11 17573.828 58454.40
2024-04-01 38014.11 14928.786 61099.44
2024-05-01 38014.11 12557.102 63471.13
2024-06-01 38014.11 10388.282 65639.95
2024-07-01 38014.11 8377.757 67650.47
2024-08-01 38014.11 6495.219 69533.01
2024-09-01 38014.11 4718.951 71309.28
2024-10-01 38014.11 3032.763 72995.47
2024-11-01 38014.11 1424.197 74604.03

Create a plot of the forecasted data points.

# Load necessary libraries
library(forecast)
library(ggplot2)
library(lubridate) # For handling date manipulations
library(kableExtra) # For creating tables

# Load the CSV file (adjust file path as needed)
file_path <- '/mnt/data/BTC-Monthly.csv'
BitCoin <- read.csv("C:/Users/chowd/Downloads/BitCoin.csv")

# Ensure the Date column is in Date format
BitCoin$Date <- as.Date(BitCoin$Date, format = "%Y-%m-%d")

# Subset the data to end in November 2023
end_date <- as.Date("2023-11-30")
BitCoin_subset <- subset(BitCoin, Date <= end_date)

# Create the time series object with monthly frequency
BitCoin_ts <- ts(BitCoin_subset$Close, start = c(2015, 1), frequency = 12)

# Fit the ARIMA(0,1,1) model
final_model <- forecast::Arima(BitCoin_ts, order = c(0, 1, 1))

# Forecast the next 12 months from December 2023 to November 2024
forecast_12_months <- forecast::forecast(final_model, h = 12)

# Create a sequence of dates for the forecast period from December 2023 to November 2024
forecast_dates <- seq(from = as.Date("2023-12-01"), by = "month", length.out = 12)

# Create a data frame for the forecast with dates, forecasted values, lower and upper bounds
forecast_df <- data.frame(
  Date = forecast_dates,
  Value = as.numeric(forecast_12_months$mean),
  Lower = forecast_12_months$lower[,2],
  Upper = forecast_12_months$upper[,2]
)

# Display the forecasted values in a table using kableExtra
forecast_table <- forecast_df %>%
  kable() %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed", "responsive"))

# Print the styled table
print(forecast_table)
## <table class="table table-striped table-hover table-condensed table-responsive" style="margin-left: auto; margin-right: auto;">
##  <thead>
##   <tr>
##    <th style="text-align:left;"> Date </th>
##    <th style="text-align:right;"> Value </th>
##    <th style="text-align:right;"> Lower </th>
##    <th style="text-align:right;"> Upper </th>
##   </tr>
##  </thead>
## <tbody>
##   <tr>
##    <td style="text-align:left;"> 2023-12-01 </td>
##    <td style="text-align:right;"> 38014.11 </td>
##    <td style="text-align:right;"> 29503.884 </td>
##    <td style="text-align:right;"> 46524.35 </td>
##   </tr>
##   <tr>
##    <td style="text-align:left;"> 2024-01-01 </td>
##    <td style="text-align:right;"> 38014.11 </td>
##    <td style="text-align:right;"> 24319.184 </td>
##    <td style="text-align:right;"> 51709.05 </td>
##   </tr>
##   <tr>
##    <td style="text-align:left;"> 2024-02-01 </td>
##    <td style="text-align:right;"> 38014.11 </td>
##    <td style="text-align:right;"> 20616.465 </td>
##    <td style="text-align:right;"> 55411.76 </td>
##   </tr>
##   <tr>
##    <td style="text-align:left;"> 2024-03-01 </td>
##    <td style="text-align:right;"> 38014.11 </td>
##    <td style="text-align:right;"> 17573.828 </td>
##    <td style="text-align:right;"> 58454.40 </td>
##   </tr>
##   <tr>
##    <td style="text-align:left;"> 2024-04-01 </td>
##    <td style="text-align:right;"> 38014.11 </td>
##    <td style="text-align:right;"> 14928.786 </td>
##    <td style="text-align:right;"> 61099.44 </td>
##   </tr>
##   <tr>
##    <td style="text-align:left;"> 2024-05-01 </td>
##    <td style="text-align:right;"> 38014.11 </td>
##    <td style="text-align:right;"> 12557.102 </td>
##    <td style="text-align:right;"> 63471.13 </td>
##   </tr>
##   <tr>
##    <td style="text-align:left;"> 2024-06-01 </td>
##    <td style="text-align:right;"> 38014.11 </td>
##    <td style="text-align:right;"> 10388.282 </td>
##    <td style="text-align:right;"> 65639.95 </td>
##   </tr>
##   <tr>
##    <td style="text-align:left;"> 2024-07-01 </td>
##    <td style="text-align:right;"> 38014.11 </td>
##    <td style="text-align:right;"> 8377.757 </td>
##    <td style="text-align:right;"> 67650.47 </td>
##   </tr>
##   <tr>
##    <td style="text-align:left;"> 2024-08-01 </td>
##    <td style="text-align:right;"> 38014.11 </td>
##    <td style="text-align:right;"> 6495.219 </td>
##    <td style="text-align:right;"> 69533.01 </td>
##   </tr>
##   <tr>
##    <td style="text-align:left;"> 2024-09-01 </td>
##    <td style="text-align:right;"> 38014.11 </td>
##    <td style="text-align:right;"> 4718.951 </td>
##    <td style="text-align:right;"> 71309.28 </td>
##   </tr>
##   <tr>
##    <td style="text-align:left;"> 2024-10-01 </td>
##    <td style="text-align:right;"> 38014.11 </td>
##    <td style="text-align:right;"> 3032.763 </td>
##    <td style="text-align:right;"> 72995.47 </td>
##   </tr>
##   <tr>
##    <td style="text-align:left;"> 2024-11-01 </td>
##    <td style="text-align:right;"> 38014.11 </td>
##    <td style="text-align:right;"> 1424.197 </td>
##    <td style="text-align:right;"> 74604.03 </td>
##   </tr>
## </tbody>
## </table>
forecast_plot <- ggplot(data = forecast_df, aes(x = Date)) +
  geom_line(aes(y = Value), color = "blue") +  # Line plot for forecasted values
  geom_point(aes(y = Value), color = "red", size = 3) +  # Points at forecasted values
  geom_ribbon(aes(ymin = Lower, ymax = Upper), fill = "blue", alpha = 0.2) +  # Confidence interval
  ggtitle("12-Month Bitcoin Price Forecast") +  # Title of the plot
  xlab("Time") + ylab("Bitcoin Price") +  # Labels for x-axis and y-axis
  scale_x_date(date_breaks = "1 month", date_labels = "%b-%Y", limits = c(min(forecast_dates), max(forecast_dates))) +  # Format x-axis with monthly intervals
  theme(axis.text.x = element_text(angle = 45, hjust = 1))

# Save the plot with increased dimensions
ggsave("Bitcoin_Forecast_Plot.png", plot = forecast_plot, width = 12, height = 6, units = "in", dpi = 300)

# Print the plot
print(forecast_plot)

4.6 Conclusion

While choosing the best model for forecasting Bitcoin prices, we evaluated the linear, quadratic, and ARIMA models, ultimately selecting the ARIMA model for its superior performance. The ARIMA model excels due to its adaptability, accurately capturing the complex and volatile nature of Bitcoin prices by integrating autoregressive and moving average components. Unlike the simpler linear and quadratic models, which struggled with Bitcoin’s non-linear trends and high volatility, the ARIMA model demonstrated robustness and accuracy, making it the optimal choice for reliable Bitcoin price forecasting in the unpredictable cryptocurrency market.